Textbook Question
"Answer the following questions for the sampling distribution of the sample mean shown to the right.
a. What is the value of μx̄?"
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7. Sampling Distributions & Confidence Intervals: Mean
Sampling Distribution of the Sample Mean and Central Limit Theorem
1:42 minutesProblem 8.1.2
Textbook Question
Suppose a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x̄ has mean μx̄ = _________ and standard deviation σx̄ = ________.
Verified step by step guidance1
Recall that the sampling distribution of the sample mean \( \overline{x} \) describes the distribution of the means of all possible samples of size \( n \) drawn from the population.
The mean of the sampling distribution of \( \overline{x} \), denoted as \( \mu_{\overline{x}} \), is equal to the population mean \( \mu \). So, \( \mu_{\overline{x}} = \mu \).
The standard deviation of the sampling distribution of \( \overline{x} \), called the standard error, is the population standard deviation \( \sigma \) divided by the square root of the sample size \( n \). This is expressed as \( \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \).
These results come from the Central Limit Theorem and properties of expected values and variances for sample means.
Therefore, to complete the blanks, you substitute the formulas: \( \mu_{\overline{x}} = \mu \) and \( \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sampling Distribution of the Sample Mean
The sampling distribution of the sample mean describes the probability distribution of the means of all possible samples of a given size from a population. It shows how sample means vary around the population mean and is fundamental for making inferences about the population.
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Sampling Distribution of Sample Mean
Mean of the Sampling Distribution (Expected Value)
The mean of the sampling distribution of the sample mean is equal to the population mean (μ). This means that on average, the sample mean is an unbiased estimator of the population mean.
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Standard Deviation of the Sampling Distribution (Standard Error)
The standard deviation of the sampling distribution, called the standard error, is equal to the population standard deviation (σ) divided by the square root of the sample size (n). It measures the variability of sample means around the population mean.
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